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3159811edo: Difference between revisions

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{{Niche}}{{clear}}
{{Mathematical interest}}{{clear}}
{{Infobox ET|Consistency=65|Distinct consistency=65}}
{{Infobox ET|Consistency=65|Distinct consistency=65}}
{{ED intro}}
{{ED intro}}


3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest EDO which is purely consistent{{idio}} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the [[harmonic series]]).
== Theory ==
Although its step size is far smaller than the human melodic [[just-noticeable difference]], 3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest edo which is purely consistent{{idio}} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the [[harmonic series]]).
 
While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is finite, and the sequence of intervals may eventually start to deviate from their true JI counterparts.


== Theory ==
=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=7|title=Approximation of odd harmonics in 3159811edo (3–15)}}
{{Harmonics in equal|3159811|columns=9}}
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=8|start=8|collapsed=true|title=Approximation of odd harmonics in 3159811edo (17–31)}}
{{Harmonics in equal|3159811|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 3159811edo (continued)}}
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=8|start=16|collapsed=true|title=Approximation of odd harmonics in 3159811edo (33–47)}}
{{Harmonics in equal|3159811|intervals=odd|prec=8|columns=8|start=24|collapsed=true|title=Approximation of odd harmonics in 3159811edo (49–63)}}


== Scales ==
== Scales ==
=== Harmonic scales ===
=== Harmonic scales ===
3159811edo accurately approximates the mode 32 of [[harmonic series]]. All interval pairs are distinguished.
As mentioned, 3159811edo accurately approximates [[32afdo|mode 32]] of the [[harmonic series]]. Additionally, unlike in [[10edo]]'s approximation of [[4afdo|mode 4]], [[87edo]]'s approximation of [[8afdo|mode 8]], or [[311edo]]'s approximation of [[16afdo|mode 16]], all interval pairs are distinguished.


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 27: Line 27:
! 38
! 38
! 39
! 39
! 40
|-
|-
! JI Ratios
! JI ratios
| 1/1
| 1/1
| 33/32
| 33/32
Line 37: Line 38:
| 19/16
| 19/16
| 39/32
| 39/32
| 5/4
|-
|-
! … in cents
! …in cents
| 0
| 0
| 53.273
| 53.273
| 104.955
| 104.955
| 155.14
| 155.140
| 203.91
| 203.910
| 251.344
| 251.344
| 297.513
| 297.513
| 342.483
| 342.483
| 386.314
|-
|-
! Degrees in 3159811edo
! Degrees in 3159811edo
Line 57: Line 60:
| 783404
| 783404
| 901817
| 901817
| 1017232
|}
|}


Line 62: Line 66:
|-
|-
! Overtones
! Overtones
! 40
! 41
! 41
! 42
! 42
Line 70: Line 73:
! 46
! 46
! 47
! 47
! 48
|-
|-
! JI Ratios
! JI ratios
| 5/4
| 41/32
| 41/32
| 21/16
| 21/16
Line 80: Line 83:
| 23/16
| 23/16
| 47/32
| 47/32
| 3/2
|-
|-
! … in cents
! …in cents
| 386.314
| 429.062
| 429.062
| 470.781
| 470.781
Line 90: Line 93:
| 628.274
| 628.274
| 665.507
| 665.507
| 701.955
|-
|-
! Degrees in 3159811edo
! Degrees in 3159811edo
| 1017232
| 1129797
| 1129797
| 1239649
| 1239649
Line 100: Line 103:
| 1654357
| 1654357
| 1752396
| 1752396
| 1848371
|}
|}


Line 105: Line 109:
|-
|-
! Overtones
! Overtones
! 48
! 49
! 49
! 50
! 50
Line 113: Line 116:
! 54
! 54
! 55
! 55
! 56
|-
|-
! JI Ratios
! JI ratios
| 3/2
| 49/32
| 49/32
| 25/16
| 25/16
Line 123: Line 126:
| 27/16
| 27/16
| 55/32
| 55/32
| 7/4
|-
|-
! … in cents
! …in cents
| 701.955
| 737.652
| 737.652
| 772.627
| 772.627
| 806.91
| 806.910
| 840.528
| 840.528
| 873.505
| 873.505
| 905.865
| 905.865
| 937.632
| 937.632
| 968.826
|-
|-
! Degrees in 3159811edo
! Degrees in 3159811edo
| 1848371
| 1942367
| 1942367
| 2034464
| 2034464
Line 143: Line 146:
| 2385302
| 2385302
| 2468949
| 2468949
| 2551089
|}
|}


Line 148: Line 152:
|-
|-
! Overtones
! Overtones
! 56
! 57
! 57
! 58
! 58
Line 158: Line 161:
! 64
! 64
|-
|-
! JI Ratios
! JI ratios
| 7/4
| 57/32
| 57/32
| 29/16
| 29/16
Line 169: Line 171:
| 2/1
| 2/1
|-
|-
! … in cents
! …in cents
| 968.826
| 999.468
| 999.468
| 1029.577
| 1029.577
Line 181: Line 182:
|-
|-
! Degrees in 3159811edo
! Degrees in 3159811edo
| 2551089
| 2631775
| 2631775
| 2711058
| 2711058
Line 192: Line 192:
|}
|}


* The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.
The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.
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